Derivations¶
Explicit formulae have been derived for the Smoluchowski equation in isotropic media for different geometries and for its logarithmically transformed version (concentration, time, space).
Law of mass action¶
The well established law of mass action give the rate of change of a concentration \(c_i\) of species \(i\):
\(t\) is time, \(c_i\) is the concentration of species \(i\), \(S_{il}\) is the net stoichiometric coefficient of species \(i\) in reaction \(l\) and \(r_l\) is the rate of reaction \(l\), which for a mass-action type of rate law is given by:
where \(\kappa_l\) is the rate constant, \(R_{kl}\) is the stoichiometric coefficient of species \(k\) on the reactant side.
If we introduce the logarithmically transormed concentration \(z\):
we have:
which can be expressed in \(z_i\):
where we may now express \(r_l\) as:
Diffusion equation¶
Laplace operator¶
Jacobian elements¶
Boundary conditions¶
Two types of boundary reflections are supported (both of Robin type): reflective and interpolating. Depending on the situation the user may want to combine these, e.g. using a reflective boundary condition close to a particle surface and interpolating BC on the other end to simulate an infinite volume.
Finite difference scheme¶
The finite difference scheme employed is that of Fornberg, which allows us to genereate finite difference weights for an arbitrarily spaced grid. This is useful for optimizing the grid spacing by using a finer spacing close to e.g. reactive surfaces.